Nonnegativity of CR Paneitz Operator and Its Application to the CR Obata’s Theorem

Abstract

In this paper, we first prove the CR analogue of M. Obata's theorem on a closed pseudohermitian (2n + 1)-manifold with free pseudohermitian torsion. Secondly, we have the CR analogue of Li-Yau's eigenvalue estimate on the lower bound estimate of positive first eigenvalue of the sub-Laplacian on a closed pseudohermitian (2n + 1)-manifold with a more general curvature condition for n ≥ 2. The key step is a discovery of CR analogue of Bochner formula which involving the CR Paneitz operator and nonnegativity of CR Paneitz operator P 0 for n ≥ 2.

Uniformization of spherical CR manifolds

“…The main difference between our paper and [22] is the treatment of the CR Bochner formula. We have realized the important role of that Paneitz-like operator P in the CR setting of the Bochner formula through the study of some other problems in recent years (e.g., [4], [5]). So we can clarify some estimates in [22] and conclude a new result in the case of n = 2.…”

“…First observe that the Paneitz-like operator P is nonnegative for ϕ ∈ C ∞ 0 ( M ) in 2.4 if n ≥ 2 (Extending Theorem 3.2 in [5] to this situation). With respect toθ (Heisenberg flat), the torsion vanishes and hence κ = 0 in (2.6).…”

The Bochner-type formula and the first eigenvalue of the sub-Laplacian on a contact Riemannian manifold

“…When Q ≡ 0, Theorems 1.1 and 1.2 coincide with the CR Bochner-type formula and CR Lichnerowiecz theorem, respectively (see e.g. [5,[9][10][11]16]). It is quite interesting to characterize the equality case of (1.6).…”

Abelian Hopf Galois structures on prime-power Galois field extensions

“…Unlike n = 1, let (M, J, θ) be a closed pseudohermitian (2n + 1)-manifold with n ≥ 2. The corresponding CR Paneitz operator is always nonnegative ( [CCC], [CC2]). …”